multiple time scale dynamics with two fast variables and one slow ...

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has a homoclinic orbit x0(t). We are interested in perturbations with s>0 and note that in this case the divergence of (3.10) is s. Hence the vector field is area expanding everywhere. The homoclinic orbit breaks for s > 0 and no periodic orbits are created. Note that this scenario does not apply to the full three-dimensional system as the equilibrium q has a pair of complex conjugate eigenvalues so that a Shil’nikov scenario can occur. This illustrates that the singular limit can be used to help locate homoclinic orbits of the full system, but that some characteristics of these orbits change in the singular limit. We are interested next in finding curves in ( ¯p, s)-parameter space that rep- resent heteroclinic connections of the fast subsystem. The main motivation is the decomposition of trajectories in the full system into slow and fast segments. Concatenating fast heteroclinic segments and slow flow segments can yield homoclinic orbits of the full system [62, 15, 69, 82]. We describe a numerical strat- egy to detect heteroclinic connections in the fast subsystem and continue them in parameter space. Suppose that ¯p∈( ¯pl, ¯pr) so that (3.10) has three hyperbolic equilibrium points xl, xm and xr. We denote by W u (xl) the unstable and by W s (xl) the stable manifold of xl. The same notation is also used for xr and tangent spaces to W s (.) and W u (.) are denoted by T s (.) and T u (.). Recall that xm is a source and shall not be of interest to us for now. Define the cross sectionΣby Σ={(x1, x2)∈R 2 : x1= xl+xr }. 2 We use forward integration of initial conditions in T u (xl) and backward integration of initial conditions in T s (xr) to obtain trajectoriesγ + andγ − respectively. We calculate their intersection withΣand define γl( ¯p, s) :=γ + ∩Σ, γr( ¯p, s) :=γ − ∩Σ 71
We compute the functionsγl andγr for different parameter values of ( ¯p, s) numerically. Heteroclinic connections occur at zeros of the function h( ¯p, s) :=γl( ¯p, s)−γr( ¯p, s) Once we find a parameter pair ( ¯p0, s0) such that h( ¯p0, s0)=0, these parameters can be continued along a curve of heteroclinic connections in ( ¯p, s) parameter space by solving the root-finding problem h( ¯p0+δ1, s0+δ2)=0 for eitherδ1 orδ2 fixed and small. We use this method later for different fixed values of y to compute heteroclinic connections in the fast subsystem in (p, s) parameter space. The results of these computations are illustrated in Figure 3.3. There are two distinct branches in Figure 3.3. The branches are asymptotic to ¯pl and ¯pr and approximately form a “V”. From Figure 3.3 we conjecture that there exists a double heteroclinic orbit for ¯p≈−0.0622. s 1.5 1 0.5 0 −0.15 −0.1 −0.05 0 0.05 ¯p Figure 3.3: Heteroclinic connections for equation (3.10) in parameter space. Remarks: If we fix p=0 our initial change of variable becomes−y= ¯p and our results for heteroclinic connections are for the FitzHugh-Nagumo equation without an applied current. In this situation it has been shown that the heteroclinic connections of the fast subsystem can be used to prove the existence of 72
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MULTIPLE TIME SCALE DYNAMICS WITH T
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MULTIPLE TIME SCALE DYNAMICS WITH T
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To my family. iv
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ing mechanical systems.”, [E.T. P
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TABLE OF CONTENTS Biographical Sket
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LIST OF TABLES 5.1 Euclidean distan
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3.1 Bifurcation diagram of (3.6). H
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4.6 Boundary conditions are blue an
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6.3 For all computations of the ful
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where (x, y)∈R m × R n are varia
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the general definition of normal hy
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fast-slow system and yields: ˙x=(D
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whereΛ,Γare matrix-valued functio
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forǫ> 0. Therefore they have no im
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The Hopf bifurcation is supercritic
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where the algorithm can be used to
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slow flow. In particular the manifo
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f ast whereφ t is the flow of the
Page 37 and 38: whereµ∈R p are parameters. Usual
Page 39 and 40: We shall not prove this result but
Page 41 and 42: d as given in equation (2.5). Let p
Page 43 and 44: variables x1= u, x2= v and y=w we g
Page 45 and 46: The geometry of the system for one
Page 47 and 48: (0, 0, 0). 2.3 The Exchange Lemma I
Page 49 and 50: withδ>0 sufficiently small. The sa
Page 51 and 52: Theorem 2.3.2. Let ¯q ∈ M∩{|a|
Page 53 and 54: exit point of a trajectory starting
Page 55 and 56: Hence we have k=1=l and n=2 in view
Page 57 and 58: The variational equations describin
Page 59 and 60: (iii) Denote byη1i the i-th row of
Page 61 and 62: Then let H(Z, X, t) := X ′ − BX
Page 63 and 64: Proof. First, we work near q, then|
Page 65 and 66: This result can be written more com
Page 67 and 68: Proof. (of Theorem 2.4.2) The argum
Page 69 and 70: ˆZ is still exponentially small. P
Page 71 and 72: in the FitzHugh-Nagumo equation ǫ
Page 73 and 74: center-unstable manifold W cu (0, 0
Page 75 and 76: where the second “fast jump” oc
Page 77 and 78: CHAPTER 3 PAPER I: “HOMOCLINIC OR
Page 79 and 80: [39, 40, 41, 42]). It states that f
Page 81 and 82: papers. Geometric singular perturba
Page 83 and 84: s 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0
Page 85 and 86: One can view this as a projection o
Page 87: for x1. We find that there are thre
Page 91 and 92: 3.3.3 Two Slow Variables, One Fast
Page 93 and 94: Also recall that the y-nullcline pa
Page 95 and 96: is negative for all values h∈(0,
Page 97 and 98: 3.4 The Full System 3.4.1 Hopf Bifu
Page 99 and 100: check whether this observation of a
Page 101 and 102: indistinguishable numerically from
Page 103 and 104: we should view this curve as an app
Page 105 and 106: s 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 sin
Page 107 and 108: waves” (see e.g. [63, 15, 69]). 3
Page 109 and 110: the singular limits but it cannot b
Page 111 and 112: Then (3.24) transforms to a three t
Page 113 and 114: with x∈R m the vector of fast var
Page 115 and 116: ons that was used by David Terman i
Page 117 and 118: 0.15 0.1 0.05 0 −0.05 0.15 0.1 0.
Page 119 and 120: from the input data. (If b−a is a
Page 121 and 122: This explicit solution provides a b
Page 123 and 124: 4.4.2 Traveling Waves of the FitzHu
Page 125 and 126: computing the homoclinic orbit, eve
Page 127 and 128: y 0.15 0.1 0.05 0 −0.05 ε=0.001,
Page 129 and 130: eturns directly to q. Figure 4.5(b)
Page 131 and 132: are complicated [107]: we expect th
Page 133 and 134: t∈[0, 1]: z ′ = T F(z) H0(z(0))
Page 135 and 136: CHAPTER 5 PAPER III: “HOMOCLINIC
Page 137 and 138: s 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 H
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For a point p∈C0 we say that C0 i
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conjugate pair of eigenvalues for t
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Proof. (Sketch) The Lyapunov coeffi
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Table 5.1: Euclidean distance in (p
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gency of W s (q) with E u (Cl,ǫ).
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y x2 W s (q) C0 Cl,ǫ x1 W s (Cl,ǫ
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periodic orbit and q are restricted
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yΣ1=ψ(Σ1) andΣ2=ψ(Σ2); see Fi
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eigenvalues, Shilnikov [102] proved
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s 1.39 1.385 1.38 1.375 1.37 1.365
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Figure 5.9(a)-(b). It was obtained
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was carried out with the stiff solv
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6.2 Introduction Our framework in t
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are called fold points. We can use
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curve C= Cl∪{(0, 0)}∪ Cr where
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explicitly. Note that currently no
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A slight modification of the formul
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6.5 Relating l1 and K Krupa and Szm
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nates. The computer algebra system
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atλ=λH= 0 forǫ= 0.05 is l MC 1
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imum and maximum of the cubic. The
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6.8 Additions It is important to no
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−7.85 −7.95 −8.05 λ x −7.9
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BIBLIOGRAPHY [1] V.I. Arnold. Encyc
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[26] M. Desroches, B. Krauskopf, an
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icated to Floris Takens. Eds.: Henk
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[74] T.J. Kaper and C.K.R.T. Jones.
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[98] H.G. Rotstein, M. Wechselberge
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